↑

Properties

Label 18.0.12614469819...8499.8
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 19^{9}$
Root discriminant $69.01$
Ramified primes $3, 7, 19$
Class number $10044$ (GRH)
Class group $[18, 558]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![77095313, -70039875, 89390700, -56155537, 40437777, -19346514, 9988128, -3865668, 1574124, -511402, 170676, -46452, 12898, -3012, 804, -196, 51, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 51*x^16 - 196*x^15 + 804*x^14 - 3012*x^13 + 12898*x^12 - 46452*x^11 + 170676*x^10 - 511402*x^9 + 1574124*x^8 - 3865668*x^7 + 9988128*x^6 - 19346514*x^5 + 40437777*x^4 - 56155537*x^3 + 89390700*x^2 - 70039875*x + 77095313)
 
gp: K = bnfinit(x^18 - 9*x^17 + 51*x^16 - 196*x^15 + 804*x^14 - 3012*x^13 + 12898*x^12 - 46452*x^11 + 170676*x^10 - 511402*x^9 + 1574124*x^8 - 3865668*x^7 + 9988128*x^6 - 19346514*x^5 + 40437777*x^4 - 56155537*x^3 + 89390700*x^2 - 70039875*x + 77095313, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 51 x^{16} - 196 x^{15} + 804 x^{14} - 3012 x^{13} + 12898 x^{12} - 46452 x^{11} + 170676 x^{10} - 511402 x^{9} + 1574124 x^{8} - 3865668 x^{7} + 9988128 x^{6} - 19346514 x^{5} + 40437777 x^{4} - 56155537 x^{3} + 89390700 x^{2} - 70039875 x + 77095313 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1261446981901370189638661902928499=-\,3^{24}\cdot 7^{12}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.01$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1197=3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1197}(1,·)$, $\chi_{1197}(835,·)$, $\chi_{1197}(457,·)$, $\chi_{1197}(778,·)$, $\chi_{1197}(400,·)$, $\chi_{1197}(151,·)$, $\chi_{1197}(856,·)$, $\chi_{1197}(1177,·)$, $\chi_{1197}(799,·)$, $\chi_{1197}(571,·)$, $\chi_{1197}(37,·)$, $\chi_{1197}(550,·)$, $\chi_{1197}(172,·)$, $\chi_{1197}(436,·)$, $\chi_{1197}(949,·)$, $\chi_{1197}(58,·)$, $\chi_{1197}(379,·)$, $\chi_{1197}(970,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{508} a^{16} - \frac{31}{127} a^{15} + \frac{39}{254} a^{14} + \frac{39}{254} a^{13} + \frac{11}{254} a^{12} + \frac{53}{254} a^{11} + \frac{1}{254} a^{10} + \frac{29}{127} a^{9} + \frac{23}{127} a^{8} + \frac{107}{254} a^{7} + \frac{12}{127} a^{6} + \frac{33}{127} a^{5} - \frac{8}{127} a^{4} - \frac{36}{127} a^{3} + \frac{45}{508} a^{2} + \frac{119}{254} a - \frac{117}{508}$, $\frac{1}{2902412340737038346235658425852935513982959040659348} a^{17} + \frac{1905131158961641055851292248284231599579716008737}{2902412340737038346235658425852935513982959040659348} a^{16} - \frac{24026502299369334101719598454472014311409849754813}{1451206170368519173117829212926467756991479520329674} a^{15} - \frac{361562843422124445174748772124738350020507567835437}{1451206170368519173117829212926467756991479520329674} a^{14} - \frac{133679979146441519996253403278162714120651677911711}{1451206170368519173117829212926467756991479520329674} a^{13} + \frac{204734817813492508546797028944704686397392492203445}{1451206170368519173117829212926467756991479520329674} a^{12} + \frac{142006197157522982535785605936066698739628682228717}{725603085184259586558914606463233878495739760164837} a^{11} - \frac{68095902288577737022509045648855665475612830211519}{725603085184259586558914606463233878495739760164837} a^{10} + \frac{175377309695775999522395548824389090070383129969458}{725603085184259586558914606463233878495739760164837} a^{9} - \frac{69719414640016487003115552572644988665461565210577}{1451206170368519173117829212926467756991479520329674} a^{8} - \frac{112205742420696007349331525695364849654087060767035}{725603085184259586558914606463233878495739760164837} a^{7} + \frac{3129131944287105130849534420293555135633744821766}{725603085184259586558914606463233878495739760164837} a^{6} - \frac{108935058609520136156433434468265076352441535005089}{725603085184259586558914606463233878495739760164837} a^{5} + \frac{4927773084852008517675156011407907951633368104869}{1451206170368519173117829212926467756991479520329674} a^{4} - \frac{244610819968631176307407382077198559750458336955225}{2902412340737038346235658425852935513982959040659348} a^{3} - \frac{273736394399963752314837487198499182313431945700507}{2902412340737038346235658425852935513982959040659348} a^{2} + \frac{545170002819941708158319379997088749170371371361517}{2902412340737038346235658425852935513982959040659348} a + \frac{145782940662230496408081078675972017199549475692635}{2902412340737038346235658425852935513982959040659348}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{18}\times C_{558}$, which has order $10044$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 54408.4888887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-19}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.45001899.1, 6.0.108049559499.8, 6.0.16468459.1, 6.0.108049559499.7, 9.9.62523502209.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$